Existence of $K$-multimagic squares and magic squares of $k$th powers with distinct entries

TL;DR

This paper advances the theory of magic squares by proving the existence of $K$-multimagic squares of order $N$ with all entries distinct whenever $N>2K(K+1)$, using a circle-method framework together with a matrix-dominance approach to control distinct-entry solutions. It also gives a direct method to produce $N\times N$ magic squares of distinct $k$th powers under explicit size bounds $N>2^{k+1}$ for $2\le k\le 4$ and $N>2\lceil k(\log k+4.20032)\rceil$ for $k\ge5$, improving on Rome and Yamagishi. A unifying counting principle for diagonal additive systems with distinct entries yields asymptotic counts $M^*_{K,N}(P)\sim cP^{N(N-K(K+1))}$ for MMS$(K,N)$ and $\#S^*_k(P;C)\sim cP^{s-rk}$ in the kth-power setting, conditional on domination and non-singular local solubility. Together, these results yield infinite families of distinct-entry squares and connect domination of coefficient matrices with partitionable submatrices via Low's work.

Abstract

We demonstrate the existence of $K$-multimagic squares of order $N$ consisting of distinct integers whenever $N>2 K(K+1)$. This improves upon our earlier result in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct $k$ th powers when $$ N> \begin{cases}2^{k+1} & \text { if } 2 \leqslant k \leqslant 4 \\ 2\lceil k(\log k+4.20032)\rceil & \text { if } k \geqslant 5\end{cases} $$ improving on a recent result by Rome and Yamagishi.

Figures (1)

• Figure 1: Best known results for $K$-multimagic squares.

Theorems & Definitions (12)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• Theorem 3.2